Given a smooth vector field $X\in\chi(M)$ on a smooth manifold $M$, I know that if I define a smooth positive function $\Phi:M\rightarrow \mathbb{R}$, then the vector field $V=\Phi X$ is just a time reparametrization of $X$.
This means that the trajectories of the two vector fields do coincide, while their orbits don't because I move along trajectories with different speeds. My question is: is there a specific expression for the time reparametrization? Can I define this reparametrization with some function like $\tau = f(t)$ related to $\Phi$ where $t$ is the old time parameter (the one used to describe the orbits of $X$) and $\tau$ the new one?
Let's be more detailed, if $\Phi_t^X(P_0)$ is an orbit of $X$, then I get that $\Phi_{\tau = f(t)}^V(P_0)$ is an orbit of $V$.